How do you integrate #int xsqrt(1-x^2)dx# from [0,1]?
Substitute:
By signing up, you agree to our Terms of Service and Privacy Policy
If one recognises that the outside of the sqrt is a function of the sqrt differentiated we can proceed by inspection.
By signing up, you agree to our Terms of Service and Privacy Policy
To integrate the function ( \int x \sqrt{1-x^2} , dx ) over the interval ([0,1]), you can use the trigonometric substitution method. Let's substitute ( x = \sin \theta ) and ( dx = \cos \theta , d\theta ). After substitution, the integral becomes ( \int \sin \theta \sqrt{1-\sin^2 \theta} \cos \theta , d\theta ).
Now, ( \sqrt{1-\sin^2 \theta} = \cos \theta ), so the integral simplifies to ( \int \sin^2 \theta , d\theta ) over the interval ([0, \frac{\pi}{2}]).
Using the identity ( \sin^2 \theta = \frac{1 - \cos 2\theta}{2} ), the integral further simplifies to ( \frac{1}{2} \int (1 - \cos 2\theta) , d\theta ).
Integrating term by term, you get ( \frac{1}{2} (\theta - \frac{1}{2} \sin 2\theta) + C ), where ( C ) is the constant of integration.
Now substitute back ( x = \sin \theta ) and ( dx = \cos \theta , d\theta ), and apply the limits ([0,1]). After substituting and simplifying, you'll get the final result for the definite integral over the interval ([0,1]).
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7